The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.77.5. Let $R \to S$ be a flat local homomorphism of local rings. Let $M$ be a finite $R$-module. Then $M$ is finite projective over $R$ if and only if $M \otimes _ R S$ is finite projective over $S$.

Proof. By Lemma 10.77.2 being finite projective over a local ring is the same thing as being finite free. Suppose that $M \otimes _ R S$ is a finite free $S$-module. Pick $x_1, \ldots , x_ r \in M$ whose images in $M/\mathfrak m_ RM$ form a basis over $\kappa (\mathfrak m)$. Then we see that $x_1 \otimes 1, \ldots , x_ r \otimes 1$ are a basis for $M \otimes _ R S$. This implies that the map $R^{\oplus r} \to M, (a_ i) \mapsto \sum a_ i x_ i$ becomes an isomorphism after tensoring with $S$. By faithful flatness of $R \to S$, see Lemma 10.38.17 we see that it is an isomorphism. $\square$


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